PomeTheorem 5.47.10 (Josefson).1822741C4B07 Let $E\subset \mathbb C^ n$ be pluripolar. Then there is $V\in \mathrm{PSH}(\mathbb C^ n)$ with $E\subset \{ V=-\infty \} $ and $V(z)\le \log ^+|z|$ for all $z\in \mathbb C^ n$.
5.47.9 Global poles on Euclidean spaceAE92E4DE9A4F
Let $E\subset \mathbb C^ n$ be pluripolar. By theorem 5.47.8, for every ball $D$ the set $E\cap D$ lies in the pole set of a plurisubharmonic function on $D$. To obtain a single function on $\mathbb C^ n$, we cannot simply extend such a function by zero outside the ball, since the extended function need not satisfy the submean inequality at the boundary of the ball. For instance, extending $|z|^2-1$ from the unit ball by zero gives $\min \{ |z|^2-1,0\} $. If $|a|=1$ and $0<r<2$, its values on the circle $\{ (1+r\mathrm e^{\mathrm i\theta })a\} $ are $\min \{ 2r\cos \theta +r^2,0\} $; they are nonpositive, and negative for $\cos \theta <-r/2$, so its mean over this circle is negative, whereas its value at $a$ is $0$.
Instead, we use the vanishing of relative extremal functions. It provides functions $u_ j$ with values in $[-1,0]$ that equal $-1$ on a bounded piece of $E$ and have small integral. We glue each of them to a small multiple of $\log ^+|z|-\log R_ j$, where $\log ^+t=\max \{ \log t,0\} $ and $R_ j>1$; this function is negative on the ball $B(0,R_ j)$ and vanishes on its boundary. The small integrals keep the sum of the glued functions locally integrable, while every point of $E$ lies in infinitely many of the pieces.
Proof. Step 1: the glued functions. Choose compact sets $L_1,L_2,\ldots $ whose interiors cover $\mathbb C^ n$, put $E_ m=E\cap L_ m$, and choose a sequence $(m_ j)_{j\ge 1}$ in which every positive integer occurs infinitely often. Since each $L_ m$ is bounded, we can choose radii $R_ j>1$ increasing to $\infty $ such that
\[ \rho _ j(z)=2^{-j}(\log ^+|z|-\log R_ j)\le -1\quad \text{on }L_{m_ j}; \]
in particular $L_{m_ j}$ lies in the ball $B_ j=B(0,R_ j)$. The set $E\cap B_ j$ is pluripolar in $B_ j$, so by theorem 5.47.8 the relative extremal function of its subset $E_{m_ j}$ with respect to $B_ j$ vanishes identically. The competitors of (5.47.4) for $E_{m_ j}$ in $B_ j$ therefore converge to zero almost everywhere on $B_ j$, and by bounded convergence their integrals over $B_ j$ tend to zero. Since a competitor for $E_{m_ j}$ with values in $[-1,0]$ equals $-1$ on $E_{m_ j}$, one of these competitors, which we call $u_ j$, satisfies
\[ -1\le u_ j\le 0,\qquad u_ j=-1\ \text{on }E_{m_ j}, \qquad \int _{B_ j}(-u_ j)\, \mathrm dV_{2n}<2^{-j}. \]
The function $\log ^+|z|=\max \{ \tfrac 12\log |z|^2,0\} $ is plurisubharmonic on $\mathbb C^ n$ by proposition 5.10.3, proposition 5.9.5, and so is $\rho _ j$. On $\partial B_ j$ we have $|z|=R_ j>1$, hence $\rho _ j=0$, while the upper limit of $u_ j$ at every point of $\partial B_ j$ is at most $0$. By proposition 5.9.5, the function $V_ j$ equal to $\max \{ u_ j,\rho _ j\} $ on $B_ j$ and to $\rho _ j$ on $\mathbb C^ n\setminus B_ j$ is plurisubharmonic on $\mathbb C^ n$. Since $\rho _ j\le -1=u_ j$ on $E_{m_ j}$, we have $V_ j=-1$ on $E_{m_ j}$. On $B_ j$ both $u_ j$ and $\rho _ j$ are nonpositive, so $V_ j\le 0$ there, while $V_ j=\rho _ j$ outside $B_ j$; hence
\[ \rho _ j\le V_ j\le 2^{-j}\log ^+|z|\quad \text{on }\mathbb C^ n. \]
In particular $V_ j$ is locally bounded.
Step 2: the sum. Fix $r>0$ and choose $j_0$ with $R_{j_0}>r$. For $j\ge j_0$ the ball $B(0,r)$ lies in $B_ j$, so $V_ j\le 0$ on $B(0,r)$, and since $V_ j\ge u_ j$ there,
\[ \int _{B(0,r)}(-V_ j)\, \mathrm dV_{2n}\le \int _{B_ j}(-u_ j)\, \mathrm dV_{2n}<2^{-j}. \]
Hence the partial sums of $\sum _{j\ge j_0}V_ j$ decrease on $B(0,r)$, and by monotone convergence their limit $W_ r$ satisfies $\int _{B(0,r)}(-W_ r)\, \mathrm dV_{2n}\le \sum _ j2^{-j}<\infty $. Thus $W_ r$ is not identically $-\infty $, and $W_ r\in \mathrm{PSH}(B(0,r))$ by theorem 5.10.1. At every point of $\mathbb C^ n$ all but finitely many terms of the series $V=\sum _ jV_ j$ are nonpositive, so $V$ is well defined with values in $[-\infty ,\infty )$. On $B(0,r)$ it equals $\sum _{j<j_0}V_ j+W_ r$, a sum of finitely many finite plurisubharmonic functions and a plurisubharmonic function; hence $V$ is plurisubharmonic on $B(0,r)$. As $r$ is arbitrary and plurisubharmonicity is a local property, $V\in \mathrm{PSH}(\mathbb C^ n)$.
Let $z\in E$, so that $z\in E_ m$ for some $m$. Since $m=m_ j$ for infinitely many $j$, infinitely many terms $V_ j(z)$ equal $-1$, while all terms with $R_ j>|z|$ are nonpositive. Hence $V(z)=-\infty $. Finally, summing the upper bounds of Step 1 gives $V\le \sum _ j2^{-j}\log ^+|z|=\log ^+|z|$. $\square$
Corollary 5.47.11.8EF0B17C7325 Let $X$ be a complex manifold of dimension $n$.
If $(u_ i)_{i\in I}$ is a nonempty family in $\mathrm{PSH}(X)$ that is locally uniformly bounded above and $F=\sup _ iu_ i$, then the set $\{ F<F^*\} $ is pluripolar.
If $(u_ j)$ is an increasing sequence in $\mathrm{PSH}(X)$ that is locally uniformly bounded above, then $\lim _ ju_ j=(\sup _ ju_ j)^*$ quasi-everywhere; here $(\sup _ ju_ j)^*$ is plurisubharmonic by theorem 5.10.2.
If $u_1,\ldots ,u_ q$, $1\le q\le n$, are locally bounded plurisubharmonic functions on $X$, then the product $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q$ does not charge any Borel pluripolar set. More precisely, in every chart, every pluripolar set has outer measure zero for the trace measure of this product and for the total variations of its coefficient measures.
Proof. (1) Let $x\in X$, and let $B$ be a coordinate ball around $x$, identified with an open ball in $\mathbb C^ n$. On $B$ the regularization $F^*$ is the regularization of $F|_ B$, so by corollary 7.6.2 the set $L\cap \{ F<F^*\} $ has outer capacity zero in $B$ for every compact set $L\subset B$. Exhausting $B$ by countably many compact sets, we obtain $\operatorname {Cap}_ B^*(B\cap \{ F<F^*\} )=0$. By theorem 5.47.8, $B\cap \{ F<F^*\} $ lies in the pole set of a plurisubharmonic function on $B$. Hence $\{ F<F^*\} $ is pluripolar.
(2) For an increasing sequence, $\lim _ ju_ j=\sup _ ju_ j$, so this is the special case of (1) for the family $(u_ j)$.
(3) Let $E$ be pluripolar, and work in a chart, identified with an open set $\Omega \subset \mathbb C^ n$. Let $B'\Subset B\Subset \Omega $ be concentric open balls. The set $E\cap B$ is pluripolar in $B$, so $\operatorname {Cap}_ B^*(E\cap B)=0$ by theorem 5.47.8.
Let $\mu $ be the trace measure of the product, or the total variation of one of its coefficient measures. The restrictions of $u_1,\ldots ,u_ q$ to $B$ form a family that is uniformly bounded on each compact subset of $B$, since $\overline B\subset \Omega $ is compact; so lemma 7.5.7(1), applied to this family and the compact set $\overline{B'}$, gives a constant $C$ with $\mu (A)\le C\operatorname {Cap}_ B(A)$ for every Borel set $A\subset \overline{B'}$. Given $\varepsilon >0$, choose an open set $G\supset E\cap B$ with $\operatorname {Cap}_ B(G)<\varepsilon $; then the outer measure of $E\cap B'$ satisfies $\mu ^*(E\cap B')\le \mu (G\cap B')\le C\operatorname {Cap}_ B(G\cap B')\le C\operatorname {Cap}_ B(G)<C\varepsilon $. Hence $\mu ^*(E\cap B')=0$. Since $\Omega $ is covered by countably many such balls $B'$, the set $E\cap \Omega $ has outer measure zero for $\mu $. For a Borel set $E$ this means that the product does not charge $E$. $\square$
Example 5.47.12.174FD6E0DD10 The torus $\mathbb T^2=\{ z\in \mathbb C^2:|z_1|=|z_2|=1\} $ is real polar, in the sense of section 5.10, but not pluripolar.
To see that $\mathbb T^2$ is real polar, let $\mu $ be the probability measure on $\mathbb T^2$ obtained as the image of $(2\pi )^{-2}\, \mathrm d\theta $ on $[-\pi ,\pi )^2$ under $\theta \mapsto (\mathrm e^{\mathrm i\theta _1},\mathrm e^{\mathrm i\theta _2})$, and put
\[ u(z)=-\int _{\mathbb T^2}|z-\zeta |^{-2}\, \mathrm d\mu (\zeta ). \]
If we identify $\mathbb C^2$ with $\mathbb R^4$, then $u$ is a positive multiple of the fundamental potential $E_4*\mu $, so it is subharmonic on $\mathbb R^4$ by proposition 3.4.11. It is finite off $\mathbb T^2$, since for $z\notin \mathbb T^2$ the distances $|z-\zeta |$, $\zeta \in \mathbb T^2$, are bounded below by the distance from $z$ to the compact set $\mathbb T^2$. Now let $z=(\mathrm e^{\mathrm i\alpha _1},\mathrm e^{\mathrm i\alpha _2})\in \mathbb T^2$. By rotation invariance, $\mu $ is also the image of $(2\pi )^{-2}\, \mathrm d\theta $ under $\theta \mapsto \zeta (\theta )=(\mathrm e^{\mathrm i(\alpha _1+\theta _1)},\mathrm e^{\mathrm i(\alpha _2+\theta _2)})$, and
\[ |z-\zeta (\theta )|^2=|1-\mathrm e^{\mathrm i\theta _1}|^2+|1-\mathrm e^{\mathrm i\theta _2}|^2\le |\theta |^2. \]
Hence
\[ -u(z)\ge \frac1{(2\pi )^2}\int _{|\theta |<\pi }\frac{\mathrm d\theta }{|\theta |^2} =\frac1{2\pi }\int _0^\pi \frac{\mathrm dr}{r}=\infty , \]
so $\mathbb T^2\subset \{ u=-\infty \} $, and $\mathbb T^2$ is real polar.
Suppose that $\mathbb T^2$ were pluripolar. By theorem 5.47.10 there would be $V\in \mathrm{PSH}(\mathbb C^2)$ with $\mathbb T^2\subset \{ V=-\infty \} $. Fix $c\in \mathbb C$ with $|c|=1$. By proposition 5.9.3, the function $f(\zeta )=V(\zeta ,c)$ is either identically $-\infty $ or subharmonic on $\mathbb C$, and $f=-\infty $ on the unit circle. In the second case, upper semicontinuity gives $\varlimsup _{\zeta \to \zeta _0}f(\zeta )\le f(\zeta _0)=-\infty $ at every point $\zeta _0$ of the unit circle, and the maximum principle (proposition 3.4.7), applied to $f+M$ on the unit disc for every $M>0$, shows that $f=-\infty $ on the unit disc. Hence $V=-\infty $ on $\{ |z_1|\le 1,\ |z_2|=1\} $. Now fix $z_1$ with $|z_1|<1$; the function $w\mapsto V(z_1,w)$ equals $-\infty $ on the unit circle, and the same argument shows that it equals $-\infty $ on the unit disc. Thus $V=-\infty $ on the open bidisc $\{ |z_1|<1,\ |z_2|<1\} $, contradicting the local integrability of $V$ (theorem 5.9.1).
For $n\ge 3$, the same arguments show that $\mathbb T^2\times \mathbb C^{n-2}\subset \mathbb C^ n$ is real polar but not pluripolar. The function $U(z)=u(z_1,z_2)$ is subharmonic on $\mathbb R^{2n}$, by lemma 4.2.7 with $k=4$ and $d=2n$. The function $U$ is finite off $\mathbb T^2\times \mathbb C^{n-2}$ and equal to $-\infty $ on it. If the set were pluripolar, theorem 5.47.10 would give $V\in \mathrm{PSH}(\mathbb C^ n)$ with poles on it, and the argument above, with the last $n-2$ coordinates fixed, would show that $V=-\infty $ on the product of the open bidisc with $\mathbb C^{n-2}$, which is again impossible.
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